Recently there has been a lot of progress in relating the exact time-complexities of various problems that have polynomial running times. Although there has been success in a variety of graph-theoretic, geometric and string problems [1]}, [2]}, [3]}, there are very few fine grained reductions from the assumptions therein to linear algebraic problems, an example being, the work of Musco et al [4]} that showed conditional lower bounds for spectrum approximation. We note here that there is a barrier to study the hardness of problems such as the rank-finding problem and solving linear equations under standard fine-grained complexity assumptions like SETH because these problems have linear time co-nondeterministic algorithmsFor example, for an unsatisfiable linear system \((A, b)\)  one can certify unsatisfiability by a witness \(w\)  such that \(w^TA = \mathbf {0}\)  but \(w^Tb \ne 0\) . and SETH-based hardness would falsify the NSETH conjecture [5]}.
